1. Field of the Invention
This invention relates to a method and apparatus for determining statistical parameters of a signal from a finite set of its samples, and is particularly but not exclusively applicable to characterizing and classifying physical entities, including complex man-made objects such as ground vehicles, by utilizing information contained in the fluctuating power of electromagnetic waves backscattered by such entities. The following description will mainly refer to classification of objects, but is predominantly applicable also to classification of other entities, such as sea clutter, sound waves, etc.
2. Description of the Prior Art
There are many circumstances in which an object, system or phenomenon under examination modifies some characteristics of a probing signal, be it electrical, acoustic, electromagnetic (which is intended herein to include both radio and optical signals) and the like. In one class of applications, only the intensity or power of the response to a probing signal can be determined. In general, at least some incomplete information related to the object's features and characteristics will be encapsulated in the observed response signal. Consequently, any inference on the object under examination must include a step of constructing a set of informative descriptors or parameters characterizing the response signal.
As will be described in more detail below, an example in which determining descriptors of a response signal is useful is that of ground-vehicle classification based on illuminating a vehicle of interest with pulses of electromagnetic wave energy and analysing the power of signals reflected by the vehicle. Such an all-weather classification technique can be utilized in various automated surveillance systems installed for monitoring purposes, e.g., to offer improved continuous surveillance of roads leading to bridges, tunnels and critical industrial infrastructure, including power plants, gas and oil storage systems, water storage and supply systems, etc.
A typical man-made object of interest, such as a ground vehicle, consists of a plurality of component scatterers of regular design, including flat plates, curved surfaces, corner reflectors, various cavities and the like. For a fixed frequency of the interrogating waveform and for a fixed aspect angle, the reflection from every individual scatterer can be regarded as a vector quantity, characterized by its specific magnitude and the phase angle. Consequently, the total signal backscattered by a complex man-made object results from the vector summation of the reflections contributed by every scatterer making up the object. Therefore, even very small changes in the aspect angle of a complex man-made object may produce large fluctuations in the backscattered signal.
Furthermore, if the separation between dominant component scatterers is comparable to a large number of wavelengths, then even a fractional change in carrier frequency of the interrogating waveform will drastically change the result of the vector summation, hence the value of reflected power.
In the case of ground-vehicle classification, it may be assumed that the power of backscattered signals is both frequency dependent and (aspect) angle dependent. August W. Rihaczek and Stephen J. Hershkowitz: “Theory and Practice of Radar Target Identification”, Artech House. Boston 2000, contains a detailed analysis of various scattering phenomena observed experimentally when examining complex man-made objects.
FIG. 1 depicts a hypothetical experiment in which a vehicle VH is rotated on a turntable TT. A transmitter TX utilizes suitable pulses of electromagnetic wave energy to illuminate the vehicle via a transmit antenna TA. A composite signal backscattered from the vehicle VH is captured by a receive antenna RA connected to a stationary receiver RX which incorporates a suitable analyser AN to determine the instantaneous power of the signal. Because the aspect angle ξ of the vehicle VH with respect to the receiver RX is changing continually, the level of the signal backscattered from the vehicle will fluctuate in some irregular manner. Some selected examples of scatter data obtained from turntable experiments are presented in Peyton Z. Peebles Jr.: “Radar Principles”. Wiley, New York 1998.
If the time intervals between the interrogating pulses are large enough, the corresponding pulses reflected from the vehicle will be uncorrelated. Additionally, irrespective of the intervals between the transmitted pulses, decorrelation of the reflected pulses can be achieved by exploiting the so-called frequency agility technique, i.e. by suitably shifting the value of transmitted carrier frequency from pulse to pulse. The use of uncorrelated responses is advantageous as it improves the efficiency of statistical inference procedures. The frequency-agility technique is well known to those skilled in the art.
Owing to the irregular nature of signals backscattered by complex man-made objects, the power of a signal reflected by a moving ground vehicle can be regarded as a random variable which may assume only positive values. In microwave remote sensing applications, fluctuating power reflected by complex objects is characterized by several well-known probability density functions (pdfs), including a gamma pdf of the form
      p    ⁡          (                        x          ;          σ                ,        α            )        =            α                        Γ          ⁡                      (            α            )                          ⁢        σ              ⁢                  (                              α            ⁢                                                  ⁢            x                    σ                )                    α        -        1              ⁢          exp      ⁡              (                  -                                    α              ⁢                                                          ⁢              x                        σ                          )            where x is the random power, Γ( ) is the gamma function, α is the shape parameter, and σ is the scale parameter. For α=1 and α=2, the above pdf yields two popular Swerling models, known to those skilled in the art. In the case when α=1, a gamma distribution reduces to an exponential distribution. FIG. 2 shows the probability density functions representing the two Swerling models.
Other popular statistical models of backscattered power are based on a Weibull distribution and a log-normal distribution. Yet another model, exploiting a Rice distribution, is useful when an object of interest comprises one dominant reflector and a plurality of smaller scatterers.
There also exists a broad class of statistical models of backscattered power based on an exponential distribution
      p    ⁡          (              x        ;        σ            )        =            1      σ        ⁢          exp      ⁡              (                  -                      x            σ                          )            modified in such a way that the scale parameter σ itself is a random variable. When the scale parameter is distributed according to a gamma distribution, the resulting model has the so-called K distribution. A Suzuki model for reflected power is obtained from the primary exponential model, when the scale parameter itself is a log-normal random variable.
All the above and also other stochastic models of fluctuating power reflected by complex objects depend on two parameters: one related to the mean value of reflected power and one characterizing the shape of the underlying statistical distribution. The properties and the applicability of the above statistical models of reflected power, summarised in Peyton Z. Peebles Jr.: “Radar Principles”. Wiley, New York 1998 and Fred E. Nathanson: “Radar Design Principles”, 2nd Ed., McGraw-Hill, New York 1991, are well known to those skilled in the art.
The statistical models discussed above are very useful in theoretical studies, especially on object detection, because each underlying probability density function is given in an analytical (explicit or implicit) form. However, the applicability of those models to practical problems of object classification is limited for the following reasons:                In all the proposed statistical models, when a scale parameter is fixed, only a single parameter governs the distribution shape, which includes both the main ‘body’ of the distribution as well as its tail. When the number of samples obtained experimentally is small or moderate (say, less than one thousand), any statistical inference regarding the tail of the underlying distribution cannot be reliable. Therefore, the performance of classification procedures based on a single shape parameter will be unsatisfactory.        In many cases, when a histogram (i.e., an empirical distribution) obtained from experimental data is compared to a range of model distributions to find ‘the best fit’, there will be several equally good (or equally bad) models matching the data. Obviously, such intrinsic ambiguity cannot facilitate reliable classification.        Although the number of postulated statistical models can be significant, there will still be an infinite number of other admissible distributions which may be more useful in characterizing a given set of experimental data. For example, if M1 and M2 are two primary model distributions, a new ‘randomised mixture’ model can be constructed by selecting M1 with probability η, and M2 with probability (1−η). Obviously, the resulting model cannot be represented adequately by either one of the two underlying models.        
In an attempt to characterize the shape of a model distribution, irrespective of its analytical form, various ‘shape parameters’ have been introduced. For example, it is known to utilize the mean-to-median ratio in order to measure the skewness of a distribution of fluctuating power. For the two Swerling models, when α=1 and α=2, the respective ratios equal 1.44 and 1.18. It should be pointed out that the mean-to-median ratio is independent of any scale parameter.
There are also widely used in mathematical statistics two parameters, γ1 and γ2, associated with the skewness and kurtosis (i.e., a measure of flatness) of a probability distribution. Practical estimates of the two parameters, γ1 and γ2, are based on higher-order sample moments of a distribution under consideration. If {x1, x2, . . . xN} is a set of N observations, then the estimates of γ1 and γ2 can be determined from
            γ      1        =                                                      1              N                        ⁢                                          ∑                                  i                  =                  1                                N                            ⁢                                                (                                                            x                      i                                        -                    μ                                    )                                3                                                                        [                                                1                  N                                ⁢                                                      ∑                                          i                      =                      1                                        N                                    ⁢                                                            (                                                                        x                          i                                                -                        μ                                            )                                        2                                                              ]                                      3              /              2                                      ⁢                                  ⁢                  γ          2                    =                                                  1              N                        ⁢                                          ∑                                  i                  =                  1                                N                            ⁢                                                (                                                            x                      i                                        -                    μ                                    )                                4                                                                        [                                                1                  N                                ⁢                                                      ∑                                          i                      =                      1                                        N                                    ⁢                                                            (                                                                        x                          i                                                -                        μ                                            )                                        2                                                              ]                        2                          -        3              where      μ    =                  1        N            ⁢                        ∑                      i            =            1                    N                ⁢                  x          i                    In the special case of a Gaussian distribution, γ1=0 and also γ2=0.
The parameters γ1 and γ2, or some others related to them functionally, appear frequently in the statistical literature. However, their applicability to the processing of experimental data on power reflected from complex man-made objects is limited for the three main reasons:                Samples of observed power are always non-negative, hence in general, the underlying distribution will be skewed to the right; however, such prior information is not incorporated in the skewness parameter γ1.        When the number of samples of observed power is small or moderate (say, less than one thousand), the statistical errors associated with the estimates of γ1 and γ2 will be too large for reliable object classification.        In general, statistics based on sample moments are not ‘robust’, i.e., their resulting values can be significantly influenced by a relatively small number of observations with unusually large or small values (so-called ‘outliers’).        
It would therefore be desirable to provide an improved method and an apparatus for determining informative shape descriptors of statistical distributions of randomly fluctuating power or intensity, especially for the purpose of entity classification based on the analysis of backscattered electromagnetic wave energy.